Combined uncertainty
What is uncertainty propagation?
Uncertainty propagation estimates the uncertainty in a calculated result when the input measurements each contain uncertainty.
The propagation rule depends on whether the measurements are added, subtracted, multiplied, or divided.
Addition and subtraction
Add absolute uncertainties
Propagated absolute uncertaintyΔR = ΔA + ΔB
This rule applies to both addition and subtraction because the uncertainty widths combine regardless of the operation sign.
Multiplication and division
Add relative uncertainties
Combined relative uncertaintyΔR ÷ |R| = ΔA ÷ |A| + ΔB ÷ |B|
After combining the relative uncertainties, multiply by the absolute result to obtain the propagated absolute uncertainty.
Worked example
Multiply 10 ± 0.2 by 5 ± 0.1
- Calculated value: 10 × 5 = 50.
- First relative uncertainty: 0.2 ÷ 10 = 0.02.
- Second relative uncertainty: 0.1 ÷ 5 = 0.02.
- Combined relative uncertainty = 0.04.
- Propagated absolute uncertainty: 50 × 0.04 = 2.
- Final result = 50 ± 2.
Zero results
Relative uncertainty can be undefined
Addition or subtraction can produce a result of zero while still having a valid absolute uncertainty. In that case, relative and percentage uncertainty are undefined because division by zero is not possible.
Scope
Understand the propagation model
- The calculator uses standard classroom worst-case propagation rules.
- It assumes the input uncertainties are independent estimates.
- More advanced statistical analysis may use quadrature or covariance-based methods.
- The final uncertainty should be rounded consistently with the reported value.
Continue your analysis
Report propagated uncertainty clearly
Apply the correct rounding rules with our significant figures guide, then explain the result and its limitations using the lab report discussion guide.
Questions and answers
Uncertainty propagation FAQ
How do you propagate uncertainty in addition and subtraction?
For addition and subtraction, add the absolute uncertainties of the measured values.
How do you propagate uncertainty in multiplication and division?
For multiplication and division, add the relative uncertainties, then multiply the combined relative uncertainty by the absolute result to obtain the propagated absolute uncertainty.
Can uncertainty ever be negative?
No. Absolute uncertainty represents a magnitude and must be zero or positive.
Why is relative uncertainty undefined when the result is zero?
Relative uncertainty requires division by the magnitude of the result, so it is undefined when the calculated result equals zero.